Do I need to remove the baseboards before putting down new laminate flooring?

You can leave the baseboards and install the Laminate normally if you intend to cover the gaps with quarter round. The baseboard is meant to cover the 1/4” gap. There is always the possibility of damaging your baseboards up removal. If you plan to remove them and put them back I suggest scoring the seem between the Wall and the baseboard and slowly prying along the line, do not pull all on one side at once as your baseboard is most likely nailed and will break with too much force. The proper installation method is to remove the baseboards to prevent the risk of a Gap showing after installation of quarter round, and so the connections of the Laminate are full secure. Bear in mind as well, you will need to cut the bottom of the door if your are looking to fit the Laminate underneath and for it to look seamless, in order to do this lay a plank on-top of the desired subfloor/underlayment and then cut the bottom of the door matching the planks height.